Quantitative model for predicting lymph formation and muscle compressibility in skeletal muscle during contraction and stretch Laura Causey, Stephen C. Cowin, and Sheldon Weinbaum1 Department of Biomedical Engineering, The City College of New York, New York, NY 10031 Contributed by Sheldon Weinbaum, April 20, 2012 (sent for review March 2, 2012) Skeletal muscle is widely perceived as nearly incompressible despite perimysial compartments. It is the purpose of this paper to develop the fact that blood and lymphatic vessels within the endomysial and a theoretical framework and simplified anatomical model for perimysial spaces undergo significant changes in diameter and performing just such an analysis. length during stretch and contraction. These fluid shifts between Perhaps the most extensive experimental investigation of the fascicle and interstitial compartments have proved extremely diffi- volume changes mentioned above is attributable to Mazzoni et al. cult to measure. In this paper, we propose a theoretical framework (5) who studied entire cross-sections of the rat spinotrapezius based on a space-filling hexagonal fascicle array to provide pre- muscle, a muscle comparable to a portion of the trapezius muscle in humans. These investigators made detailed measurements of dictions of the displacement of blood and lymph into and out of the fi muscle’s endomysium and perimysium during stretch and contrac- the cross-sectional area of individual muscle bers, the thickness tion. We also use this model to quantify the distribution of blood and of the interstitial (perimysial) space between fascicles, and the initial lymphatic (IL) vessels within a fascicle and its perimysial space cross-sectional area of the ILs in planes transverse to the primary using data for the rat spinotrapezius muscle. On average, there are arterioles and venules in the perimysial space. This was done for both electrically excited muscular contractions and passive stretches 11 muscle fibers, 0.4 arteriole/venule pairs, and 0.2 IL vessels per of up to 20%. Their detailed measurements of fiber cross-sections fascicle. The model predicts that the blood volume in the endomysial within the fascicles showed small, unexplained departures from space increases 24% and decreases 22% for a 20% contraction and isovolumetry. Unfortunately, it was too difficult to make meas- stretch, respectively. However, these significant changes in blood ∼ urements of the TAs and CVs within the fascicles or estimate volume in the endomysium produce a change of only 2% in fascicle their changes in diameter during contraction and stretch. These cross-sectional area. In contrast, the entire muscle deviates from investigators did make detailed measurements of the width of isovolumetry by 7% and 6% for a 20% contraction and stretch, re- the perimysial space, which was observed to be of nearly uniform spectively, largely attributable to the significantly larger blood vol- thickness, except in regions where there was a primary arteriole/ ume changes that occur in the perimysial space. This suggests that venule pair and, in some cases, an intervening IL. Most in- arcade blood vessels in the perimysial space provide the primary terestingly, they observed that the size of the ILs increased during pumping action required for the filling and emptying of ILs during stretch and decreased during active contraction. This last obser- muscular contraction and stretch. vation supported their hypothesis promulgated by Skalak et al. (3) that the dilation and contraction of the primary arterioles and dimensions | spacing | resting | isovolumetric | deformation venules served as a pump to fill and empty the ILs, which are known to be devoid of active contractile elements. Subsequently, Mazzoni n the 17th century, Jan Swammerdam placed an isolated frog’s et al. (5) proposed that the increased diameters of the muscle fibers Ithigh muscle in an airtight syringe and measured the volume of during contraction also compressed adjacent IL vessels and the the muscle as it contracted and relaxed by observing the movement decreased fiber muscle diameter during stretch also opened the ILs. PHYSIOLOGY of a droplet of water in the end of the syringe, thus inventing the They were unable to establish the relative importance of these first plethysmograph. He observed that the volume of the muscle two mechanisms, an important objective of our theoretical model, did not change as it was excited. Thus, when a muscle is isolated so because the vascular volume of the fascicle was not examined. that blood neither enters nor exits the muscle, it functions iso- There are no detailed measurements of the change in the vas- volumetrically (1). Furthermore, because the bulk modulus of cular volume within a fascicle during contraction and stretch or of water is 2.3 GPa, its resistance to volumetric deformation is very the change in IL cross-section with the dilation and constriction of large relative to its resistance to shape distortion, and muscle tissue the primary arterioles and venules. The density of the ILs has also itself is often described accurately as “incompressible.” However, never been measured, and the fraction of primary arterioles and fi ENGINEERING the entire muscle and parts of the muscle may experience volu- venules that are accompanied by an IL has not been quanti ed. In metric changes attributable to the fact that both blood and lymph the absence of such measurements, we have constructed a quan- reside in highly flexible vessels in which they can both enter and titative anatomical model for a space-filling hexagonal fascicle exit with little resistance. These fluid conduits lie in both the array interspersed with a uniformly distributed perimysial space endomysial space between individual muscle fibers within fascicles that also contains local regions of enlargement to accommodate and in the perimysial space surrounding and separating the fas- the countercurrent arcade vessels. As a substitute for much of this cicles (Fig. 1A). The endomysial space contains capillaries, ter- missing information, we have combined the measurements on minal arterioles (TAs), and collecting venules (CVs). Initial individual microvessels, subject to tetanic excitation (6, 7) and lymphatic (IL) vessels are occasionally seen in the capillary bed passive muscle length changes (8, 9), with the measurements of adjacent to muscle fibers; however, it is unclear if these vessels are the overall behavior of the fascicles and ILs in the study by involved in lymph formation and transport (2). The perimysial space contains arcade arterioles and venules (also called primary and secondary arterioles) and ILs, which lie in close proximity to Author contributions: L.C., S.C.C., and S.W. designed research, performed research, ana- the larger arterioles and venules (3, 4). Because the changes in lyzed data, and wrote the paper. blood and lymph volume between these compartments are very The authors declare no conflict of interest. small compared with the whole-muscle tissue volume, it has been 1To whom correspondence should be addressed. E-mail: [email protected]. fi dif cult to either measure these changes or quantitatively relate This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. the relative changes in volume between the endomysial and 1073/pnas.1206398109/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1206398109 PNAS | June 5, 2012 | vol. 109 | no. 23 | 9185–9190 Downloaded by guest on September 26, 2021 resting state, these investigators provided the following frac- tional area ratios. The perimysial interstitium, blood vessels, and IL vessels, are 7.4%, 11.0%, and 0.4% of the total area of the muscle, respectively. To use this information, At = Af + Ai + Ab + Al is rewritten in the following form using the area ratios to provide a correlation between Af and Ai: A A A t A ¼ A þ A þ b A þ l A A i f i A i A i i i i [1] Ai 0:11 0:004 ¼ Af þ Ai þ Ai þ Ai: 0:074 0:074 0:074 The area of the hexagonal fascicles, Af, with sides of length 2b 2 is given by Af =6√3b ,andtheareaoftheinterstitial space, Fig. 1. Muscle architecture (A) and hexagonal model (B) of muscle fascicles Ai, between the hexagons is given by Ai = 12(bh), where h is the and tissue. half-height of the perimysial space, which was previously noted to be equal to ∼3 μm (5), and b is 1/2 of the length of each hexagonal side. Substituting 6√3b2, 12(bh), and ∼3 μm into Eq. Mazzoni et al. (5) to construct a more complete picture of muscle 1, one finds that there is a single unique value of b that satisfies behavior. This includes model predictions for the frequency of the all the foregoing conditions for a space-filling hexagonal fascicle arcade vessels and their adjacent ILs and quantitative estimates of array, namely, b = 38 μm. With this result, all the unknown in- the small changes in vascular volume that occur in the endomysial dividual area components of the muscle at rest can be de- 2 2 and perimysial spaces during muscular contraction and stretch, termined. Therefore, At = 18,482 μm , Af = 15,006 μm , Ai = 2 2 2 and their relation to IL volume changes. 1,368 μm , Ab = 2,034 μm , and Al =74μm . The number of capillaries for each fascicle, ncap, was found by Anatomical Background and Model multiplying the capillary density, ρ, of 1,263 capillaries per For simplicity, we consider muscle to be composed of four hier- square millimeter (10) by the total area, At, of each muscle archical structural levels. The myofibril is the lowest level con- hexagon. We find that each muscle fascicle is paired with ∼23 sidered. It contains the force-generating sarcomeres. A number capillaries. Each TA runs transverse to the muscle fascicle di- of myofibrils are bundled into a muscle fiber, which is encapsu- rection and gives rise to capillaries.